On Some Refinements of Jensen's Inequality

Abstract

Dedicated to the Memory of Professor M. H. AfghahiThis paper contains a link between Probability Proportional to Size (PPS) sampling and interpolations of the classical Jensen inequality. We show that these interpolating inequalities are, in fact, special cases of the conditional Jensen inequality when applied over an appropriate probability space. We provide a few examples dealing with divided differences, convex functions over linear spaces, approximation operators, and sampling with and without replacement. 1998Academic Press

On the Monotonicity of Positive Linear Operators

“…In particular, [14] gave a probabilistic method of showing that, under the condition of existence of first moment, K n have to be monotone from above for convex functions. A different proof of this result and some modifications were recently provided by [13].…”